TY - JOUR
T1 - Sweeping Preconditioners for the Iterative Solution of Quasiperiodic Helmholtz Transmission Problems in Layered Media
AU - Nicholls, David P.
AU - Pérez-Arancibia, Carlos
AU - Turc, Catalin
N1 - Funding Information:
DPN gratefully acknowledges support from NSF through Contract DMS-1522548 and DMS-1813033. CT gratefully acknowledges support from NSF through Contract DMS-1614270.
Publisher Copyright:
© 2020, Springer Science+Business Media, LLC, part of Springer Nature.
PY - 2020/2/1
Y1 - 2020/2/1
N2 - We present a sweeping preconditioner for quasi-optimal domain decomposition methods (DD) applied to Helmholtz transmission problems in periodic layered media. Quasi-optimal DD (QO DD) for Helmholtz equations rely on transmission operators that are approximations of Dirichlet-to-Neumann (DtN) operators. Employing shape perturbation series, we construct approximations of DtN operators corresponding to periodic domains, which we then use as transmission operators in a non-overlapping DD framework. The Robin-to-Robin (RtR) operators that are the building blocks of DD are expressed via robust boundary integral equation formulations. We use Nyström discretizations of quasiperiodic boundary integral operators to construct high-order approximations of RtR. Based on the premise that the quasi-optimal transmission operators should act like perfect transparent boundary conditions, we construct an approximate LU factorization of the tridiagonal QO Schwarz iteration matrix associated with periodic layered media, which is then used as a double sweep preconditioner. We present a variety of numerical results that showcase the effectiveness of the sweeping preconditioners applied to QO DD for the iterative solution of Helmholtz transmission problems in periodic layered media.
AB - We present a sweeping preconditioner for quasi-optimal domain decomposition methods (DD) applied to Helmholtz transmission problems in periodic layered media. Quasi-optimal DD (QO DD) for Helmholtz equations rely on transmission operators that are approximations of Dirichlet-to-Neumann (DtN) operators. Employing shape perturbation series, we construct approximations of DtN operators corresponding to periodic domains, which we then use as transmission operators in a non-overlapping DD framework. The Robin-to-Robin (RtR) operators that are the building blocks of DD are expressed via robust boundary integral equation formulations. We use Nyström discretizations of quasiperiodic boundary integral operators to construct high-order approximations of RtR. Based on the premise that the quasi-optimal transmission operators should act like perfect transparent boundary conditions, we construct an approximate LU factorization of the tridiagonal QO Schwarz iteration matrix associated with periodic layered media, which is then used as a double sweep preconditioner. We present a variety of numerical results that showcase the effectiveness of the sweeping preconditioners applied to QO DD for the iterative solution of Helmholtz transmission problems in periodic layered media.
KW - Domain decomposition methods
KW - Helmholtz transmission problems
KW - Periodic layered media
KW - Sweeping preconditioners
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U2 - 10.1007/s10915-020-01133-z
DO - 10.1007/s10915-020-01133-z
M3 - Article
AN - SCOPUS:85079319169
SN - 0885-7474
VL - 82
JO - Journal of Scientific Computing
JF - Journal of Scientific Computing
IS - 2
M1 - 44
ER -